The decision problem of an agent is to maximize expected discounted utility: \sum_{t=0}^\infty \sum_{s^t} \beta^t \pi_t(s^t)U(c_t(s^t)) subject to the budget constraint: a_{t+1}(s^t) = R(a_t(s^{t-1}) + y_t(s^t) - c_t(s^t)) where y_t(s^t) denotes the agent's earnings following history s^t, and a_t(s^{t-1}) is the agent's wealth at the beginning of period t, following history s^{t-1}. The probability of a given history s^t is denoted by \pi_t(s^t). Here, we do not specify a particular stochastic process for s^t, so it might be Markov or not. By assumption, the agent cannot borrow, so: a_{t+1}(s^t) \ge 0. This constraint might be binding in certain states of the world. Assume that the utility function is such that U'(c) \to \infty as c \to 0, so the agent will never choose to consume zero. Set up the Lagrangian for this problem and derive the first-order conditions